1,2
instance w ∈ W
0,410 350,35T , t Uncertain transaction database, transaction
0,30,8t ∈ T
0,250,60 60,2I Set of all items
0,150,1X, x Itemset X ⊆ I, item x ∈ I
0,05S(X, w) Support of X in world w
00 1 2 3 4 5 6P i (X) Probability that the support of X is i
minimum support (minSup)
support i
P ≥i (X ) Probability that the support of X is at least i
P i,j (X) Probability that i of the first j transactions
(b) Frequentness
(a) Support proba-
contain X
bility distribution of
probabilities of {D}
P ≥i,j (X ) Probability that at least i of the first j
{D}
transactions contain X
Figure 4: Probabilistic support of itemset X = {D}
Figure 3: Summary of Notations
in the uncertain database of Figure 2.
contrast to {D}. An expected support based technique does
Note that the transaction subset S ⊆ T contains exactly i
not differentiate between the two.
transactions.
The confidence with which an itemset is frequent is very
important for interpreting uncertain itemsets. We there-
Proof. The transaction subset S ⊆ T contains i transac-
tions. The probability of a world w j where all transactions
fore require concepts that allow us to evaluate the uncertain
in S contain X and the remaining |T −S| transactions do not
data in a probabilistic way. In this section, we formally in-
contain X is P(w j ) = Q
troduce the concept of probabilistic frequent itemsets.
t∈S P (X ⊆ t) · Q
t∈T−S (1 − P(X ⊆
t)). The sum of the probabilities according to all possi-
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